No, JT65 only requires time synchronization, and it's not a super hard requirement. For me, it seems to work much better if my clock is within 1 second of official time, but other have said that even several seconds of slop would still work.
So, yes, you could get a very good clock of any type, set it properly once, and have many lifetimes' of JT65 contacts.
My point was more of an angels-on-pin sort of one: Theoretically, the only way any two clocks can get synchronized is to either use some communications method to sync them, or to bring them together and sync them. The latter is an interesting case. Let's say I have a friend on another continent that I met once on a trip, and I brought along my rubidium clock. We set our clocks together and I went home. Now I want to talk to him over JT65. Voila, no problem -- except for the fact that JT65 conversations are very limited of course. :-)
You could also probably use some mutually observable natural phenomenon as your out-of-band signal. Like the precise moment some distant star went nova. That seems fair game, too. :-)
As I think about it more, it comes down to the concept of "the time" as we understand it. "The time" is not a real thing, but a mutually agreed upon convention that is maintained through a vast and varied network of communications channels. Those channels include, radio, optical, sound ("hey, what time is it?"), physically carrying clocks, etc. But irreducibly, there must be a network for the concept to work.